The variogram

The variogram is usually calculated from the experimental observations, and describes the spatial structure of the RF. It can be shown (5) that if $x_0, x_1, \ldots, x_n$ are $n+1$ points belonging to the domain of interpolation and the coefficients $\lambda_0, \lambda_1, \ldots, \lambda_n$ satisfy:

$$\sum_{i=0}^n \lambda_i = 0$$

then $\gamma(h)$ has to satisfy:

$$- \sum_{i=0}^n\sum_{j=0}^n \lambda_i\lambda_j \gamma(x_i-x_j) \ge 0$$

and

$$\lim_{\vert h\vert\rightarrow \infty} \frac{\gamma(h)}{h^2} = 0$$

i.e. $\gamma(h)$ is tends to infinity slowlier thatn $\vert h\vert^2$ as $\vert h\vert\rightarrow \infty$.

In principle there $\gamma(h)$ could assume different behaviors also with the direction of vector $h$ (``anisotropy''), but this is in general not easily verifiable due to the limited number of data points usually available for hydrologic variables. If the experimental variogram displays anisotropy, then the intrinsic hypothesis is not verified and one has to use the so called ``universal'' kriging (2,3).

Figure 2.2: Behavior of the most commmonly used variograms: polinomial (top left); exponential (top right); gaussian (bottom left); spherical (bottom right)

The most commonly used isotropic variograms are shown in Fig. 2.2 and are of the form:

1. polinomial variogram:
   $$\gamma(h) = \omega h^\alpha \qquad\qquad\qquad 0<\alpha<2$$

2. exponential variogram:
   $$\gamma(h) = \omega \left[1-e^{-\alpha h}\right]$$

3. gaussian variogram:
   $$\gamma(h) = \omega \left[1-e^{-(\alpha h)^2}\right]$$

4. spherical variogram:
   $$\gamma(h) = \left\{ \begin{array}{cc} \frac{1}{2}\omega \left[\fr... ...a} \right)^3\right] & h \le \alpha \\ \omega & h > \alpha \end{array}\right.$$

where $\omega$ and $\alpha$ are real constants.

The variogram is estimated from the available observations in the following manner. The data points are subdivided into a prefixed number of classes based on the distances between the measurement locations. For each pair $i$ and $j$ of points and for each class calculate:

1. the number $M$ tha fall within tha class;
2. the average distance of the class;
3. the half of the mean quadratic increment
   $$\frac{1}{2}\sum(Y_i-Y_j)^2/M$$

In general the pairs are not uniformly distributed among the different classes as usually there are more pairs for the smaller distances. Thus the experimental variogram will be less meaninful as $h$ increases. A best fit procedure together with visual inspection is then used to select the most appropriate variogram and evaluate its optimal parameters.